History of mathematics · Maths EE idea · Accessible
Archimedes' bounds for π
A research question to start from
How did Archimedes use inscribed and circumscribed polygons to bound π, and how quickly does his method converge compared with modern series?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
Working through a historical method in modern notation, then measuring it.
Mathematics you would need
- Perimeters of regular polygons
- Recurrences for doubling sides
- Trigonometry
- Rates of convergence
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Derive the doubling recurrences.
- Reproduce the bounds for 96 sides.
- Compare convergence with a modern series.
Scope and difficulty
Accessible. Accessible.
Pitfalls
- Biography.
- Copying historical tables without working.
Where to start reading
Search a library catalogue or a university's open lecture notes for: Archimedes method polygons pi recurrence; harmonic geometric mean. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).